SOLID STATE PHYSICS (3C25) Solutions to problem paper 1 1 ...

(f) Why is this surprising? [2]
The body centred cubic crystal has a cubic cell for which h=1. However
the basis introduces extra planes.
3. When electrons are scattered from a crystal surface
(a) Why does the angle of incidence equal the angle of reflection? [2]
To ensure that the waves scattered off neighbouring surface atoms are
in phase.
(b) Why do you only see reflections at some angles of incidence? [2]
Because only for certain angles of incidence are the waves scattered off
neighbouring planes in phase.
(c) Why can you study only atoms near the crystal surface? [2]
Electrons cannot penetrate deep into a crystal because they are scattered
by the surface layers of atoms.
4. Consider the scattering of x-rays of wavelength λ from a face centred cubic
crystal with cubic lattice constant a.
(a) What are the allowed scattering angles when the scattering is from the
(001) planes of the crystal? (Use Bragg’s law.) [3]
2d sin θ = nλ
d = a/2
⇒ sin θ = nλ
a
(b) Using the simple cubic representation of a face centred cubic crystal,
write down the primitive translation vectors and calculate the reciprocal
lattice vectors. [3]
a = a(100) A = 2π
a (100)
b = a(010) B = 2π
a (010)
c = a(001) C = 2π
a (001)
(c) Using the Laue construction, determine the allowed scattering vectors for
scattering from (001) planes. [4]
In general, a reflection from a (001) surface implies that ∆ k is parallel
to C (the primitive reciprocal lattice vector with indices (001)). From the
Laue construction we know that scattering vectors must equal reciprocal
lattice vectors for strong reflection:
2